异质介质中的竞争和边界形成：在神经元分化中的应用
Competition and boundary formation in heterogeneous media: Application to neuronal differentiation
摘要：
我们分析了一个非齐次系统的耦合反应扩散方程代表的动态基因表达在分化的神经细胞。这一发展阶段的结果是形成了由清晰而平滑的边界分隔开的不同功能区。它通过两个基因的表达之间的竞争进行，这两个基因的表达是由化学物质的单调梯度驱动的，基因表达的产物经历局部扩散并驱动邻近细胞中的基因表达。因此，问题落在一个更一般的设置中的物种在竞争中的非均匀介质。我们发现，在任意小的扩散的限制，存在唯一的单调稳定的解决方案，分裂成两个赢家通吃的神经组织在一个精确的边界点的部分：在边界的两侧，不同的神经元类型。为了进一步描述这个边界的位置，我们使用系统的爆破并定义一个由单调梯度内的位置参数化的行波问题：精确的边界位置由空间中的唯一点给出，在该点处波的速度为零。
英文摘要：
We analyze an inhomogeneous system of coupled reaction-diffusion equations representing the dynamics of gene expression during differentiation of nerve cells. The outcome of this developmental phase is the formation of distinct functional areas separated by sharp and smooth boundaries. It proceeds through the competition between the expression of two genes whose expression is driven by monotonic gradients of chemicals, and the products of gene expression undergo local diffusion and drive gene expression in neighboring cells. The problem therefore falls in a more general setting of species in competition within a non-homogeneous medium. We show that in the limit of arbitrarily small diffusion, there exists a unique monotonic stationary solution, which splits the neural tissue into two winner-take-all parts at a precise boundary point: on both sides of the boundary, different neuronal types are present. In order to further characterize the location of this boundary, we use a blow-up of the system and define a traveling wave problem parametrized by the position within the monotonic gradient: the precise boundary location is given by the unique point in space at which the speed of the wave vanishes.
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